20 ms·

But pi does contain the finite instruction sequences for programs that generate the digits of those numbers.

by wcoenen 16y ago

sp332 16y ago
Well OK, the formula at this link will give you all the digits of pi: http://en.wikipedia.org/wiki/Numerical_approximations_of_%CF%80#BBP_formula_.28base_16.29 http://en.wikipedia.org/wiki/Numerical_approximations_of_%CF... The formula is clearly finite (on the order of 40 symbols), it does not contain "all the information in the world" and neither can any sequence it generates.
- jerf 16y ago
  Information theory is a tricky beast. Pi contains no more information than it takes to express it, so in information theory terms it doesn't contain much information at all since it can be expressed very concisely. It so happens that pi + enough information to use pi as raw material to produce the desired information can output anything you like, which some people find amazing. But really, "anything + enough information to use 'anything' as raw material to produce the desired output" works equally well, too; pi isn't actually that special either. Also, the information needed to produce your desired output from pi is on average larger than the information in the desired output in the first place, which is why the "compress your things by referencing a digit number in pi" doesn't actually work. So, pi contains all the information in the world as long as you use more information to extract it from pi, in which case all you are really doing is expressing the original information in a particularly unpleasant encoding. But it sounds cool to say pi has all this information... "how can anything with an infinite number of normalized digits consist of only a handful of bits?" asks our feeble, mathematics-impaired human minds. (By the way, this is extended agreement with the post I'm replying to, not any implicit disagreement.)

20 ms·

But pi does contain the finite instruction sequences for programs that generate the digits of those numbers.

by wcoenen 16y ago

But pi does contain the finite instruction sequences for programs that generate the digits of those numbers.

sp332 16y ago
Well OK, the formula at this link will give you all the digits of pi: http://en.wikipedia.org/wiki/Numerical_approximations_of_%CF%80#BBP_formula_.28base_16.29 http://en.wikipedia.org/wiki/Numerical_approximations_of_%CF... The formula is clearly finite (on the order of 40 symbols), it does not contain "all the information in the world" and neither can any sequence it generates.
- jerf 16y ago
  Information theory is a tricky beast. Pi contains no more information than it takes to express it, so in information theory terms it doesn't contain much information at all since it can be expressed very concisely. It so happens that pi + enough information to use pi as raw material to produce the desired information can output anything you like, which some people find amazing. But really, "anything + enough information to use 'anything' as raw material to produce the desired output" works equally well, too; pi isn't actually that special either. Also, the information needed to produce your desired output from pi is on average larger than the information in the desired output in the first place, which is why the "compress your things by referencing a digit number in pi" doesn't actually work. So, pi contains all the information in the world as long as you use more information to extract it from pi, in which case all you are really doing is expressing the original information in a particularly unpleasant encoding. But it sounds cool to say pi has all this information... "how can anything with an infinite number of normalized digits consist of only a handful of bits?" asks our feeble, mathematics-impaired human minds. (By the way, this is extended agreement with the post I'm replying to, not any implicit disagreement.)